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Full text of "Treatise On Analysis Vol-Ii"

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We have seen (13.5.10) that every function/: X -> R satisfies //#(/) ^/j*(/).
The function / is said to be integrable (with respect to ju) or ^integrable if
M*(/) and /**(/) are finite and equal* Their common value is then called the
integral of f with respect to ju, and is written /i(/), or </, /*>, or \fdfji, or

[/(*) ^M*)- Clearly every function /e 2rR(X) is integrable and its integral
is the value of \i at/, so that our notation is consistent.

An integrable function is therefore./zrate almost everywhere (13.6.4), but a
bounded function is not necessarily integrable. For example, a constant non-
zero function on R, or the continuous function 1/(1 + |jc|), is not integrable
(with respect to Lebesgue measure) (cf. (13.20)).

In view of the definitions in (13.5), we have the following criterion for

(13.7.1) (i) For a function f: X-K to be integrable, it is necessary and
sufficient that, given any e > 0, there should exist g e f and he<# such that
9 ^/^ h and /x*(A)  M*(#) =>  (or, equivalently (13.5.3), p*(h  g) g e).

(ii) Iffis integrable, there exists a decreasing sequence (hn) of functions
belonging to J and an increasing sequence (gn) of functions belonging to $f
such that gn^f^ hnfor all n and

lim /

(i) The condition implies that /t*(A) and fi*(g) are necessarily finite
(and therefore so are /**(/) and /**(/)), and that n*(f)  /**(/)

(ii) For each n there exists h'n e  and g^e^ such that g'n^f^h'n and
/*(/) ~ w"1 ^ l**(g'n) ^ K/) ^ M*(%) ^ /*(/) + w"1- The required conditions
are then satisfied by taking hn = inf(A'l5 . . . , h'n) and gn  sup(#i, . . . , g).

If /is integrable, then clearly so is any function/! equivalent to/ (13.6), and
we have f/i fi?/i = |/rf/x (which we denote also by X/))- It is now clear hw
integrability should be defined for functions defined almost everywhere in
X: such a function /is integrable if the functions equivalent to /and defined

on the whole of X are integrable. The number /*(/) is also denoted by \fd\i,
or  /(x rf/<x), or

(13.7.2)   For a mapping f:X-+Htobe integrable it is necessary and
sufficient that, given any e > 0, there exists a function u e 2rR(X) such that this defines an order relation on the set of equivalence classes (with